Quick question on compact support of $f$

Question

Let $f \in C^2(\mathbb R)$ with compact support $X \subset \mathbb R $. If $f' =\frac{df}{dx}$ has a compact support.

Is $supp(f')\subset supp(f)$ ?

Would appreciate any help

Answer 1

Yes, its true. Pick an $x\in\mathbb{R}$ with $f'(x)\neq 0$. Since $$f'(x)=\lim_{y\rightarrow x}\frac{f(x)-f(y)}{x-y}\neq 0$$ we can find a sequence $y_n\rightarrow x$ such that $f(y_n)\neq f(x)$. Hence either $f(x)\neq 0$ or $f(y_n)\neq 0$. Either way $x\in supp(f)$. Since $supp(f)$ is closed, the result follows.

Answer 2

Let $r\notin\operatorname{supp}(f)$. Then there is an open interval $I$ containing $r$ such that $f(x)=0$ for every $x\in I$. Therefore $f'(x)=0$ for every $x\in I$ and so $r\notin\operatorname{supp}(f')$.